Stringy Orbifold K-Theory

Outline Intro Stringy K–Theory Summary & Future

Stringy orbifold K-Theory

Ralph Kaufmann

Purdue University

USC, November 2009

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future

1 Introduction Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation

2 Stringy K–Theory Stringy Functors Pushing, Pulling and Obstructions Stringy K–theory and the Chern Character Theorem on Variations, Algebraic Methods and Compatibility Algebraic Aspects Links to GW Invariants

3 Summary & Future

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Orbifolds

Varieties/manifolds n n Locally like Spec(A) e.g. A , R . Usually want a “nice” variety, i.e. smooth projective variety. Or smooth structure

Deligne Mumford stacks/Orbifolds Definition of stacks technically complicated. “Nice” stacks are Deligne–Mumford stacks with projective coarse moduli space. Locally think of Spec(A)/G — G a group of local automorphisms. In the differential category, being locally modeled by n R /G is one definition of an orbifold.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Orbifolds

Varieties/manifolds n n Locally like Spec(A) e.g. A , R . Usually want a “nice” variety, i.e. smooth projective variety. Or smooth structure

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Nice Stacks/Orbifolds

Global Quotients A nice class of examples are global quotients [V /G] of a variety by a ﬁnite group action. We sometimes write (V , G).

Lie quotients Interesting stacks in the diﬀerential category are given by M/G where G is a Lie group that acts with ﬁnite stabilizers. In the algebraic category the corresponding spaces are nice, that is they are examples of DM-stacks with a projective coarse moduli space.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Nice Stacks/Orbifolds

Global Quotients A nice class of examples are global quotients [V /G] of a variety by a ﬁnite group action. We sometimes write (V , G).

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Nice Stacks/Orbifolds

Global Quotients A nice class of examples are global quotients [V /G] of a variety by a ﬁnite group action. We sometimes write (V , G).

Ralph Kaufmann Stringy orbifold K-Theory and Invariants

Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Orbifolds/Group actions

Example 1: Cone

Let Cn be the cyclic group of n–th roots of unity and let ζn be a generator. An example over C: 2πi/3 4πi/3 C3 = {1, e , e }, C4 = {1, i, −1, −i}

Cn acts by rotation.

The example of C3: Action

Ralph Kaufmann Stringy orbifold K-Theory and Invariants

Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Orbifolds/Group actions

Example 1: Cone

Let Cn be the cyclic group of n–th roots of unity and let ζn be a generator. An example over C: 2πi/3 4πi/3 C3 = {1, e , e }, C4 = {1, i, −1, −i}

Cn acts by rotation.

The example of C3: Action

Ralph Kaufmann Stringy orbifold K-Theory and Invariants

Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Orbifolds/Group actions

Example 1: Cone

Let Cn be the cyclic group of n–th roots of unity and let ζn be a generator. An example over C: 2πi/3 4πi/3 C3 = {1, e , e }, C4 = {1, i, −1, −i}

Cn acts by rotation.

The example of C3: Action

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Orbifolds/Group actions

Example 1: Cone

Let Cn be the cyclic group of n–th roots of unity and let ζn be a generator. An example over C: 2πi/3 4πi/3 C3 = {1, e , e }, C4 = {1, i, −1, −i}

Cn acts by rotation.

The example of C3: Action and Invariants

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Orbifolds

Example 2: Symmetric Products ×n X = X × · · · × X , Sn group of permutations. ×n (X , Sn), where acts via σ ∈ Sn : σ(x1,..., xn) = (xσ(1),..., xσ(n)).

A picture for n = 2 • Y = X × X • ∆ = {(x, x)|x ∈ X } ⊂ Y • τ = (12) • τ(x, y) = (y, x)

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Orbifolds

Example 2: Symmetric Products ×n X = X × · · · × X , Sn group of permutations. ×n (X , Sn), where acts via σ ∈ Sn : σ(x1,..., xn) = (xσ(1),..., xσ(n)).

A picture for n = 2 X ∆ • Y = X × X • ∆ = {(x, x)|x ∈ X } ⊂ Y (x,y) • τ = (12) • τ(x, y) = (y, x)

XxX X

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Orbifolds

Example 2: Symmetric Products ×n X = X × · · · × X , Sn group of permutations. ×n (X , Sn), where acts via σ ∈ Sn : σ(x1,..., xn) = (xσ(1),..., xσ(n)).

A picture for n = 2 X ∆ • Y = X × X • ∆ = {(x, x)|x ∈ X } ⊂ Y (x,y) • τ = (12) (y,x) • τ(x, y) = (y, x)

XxX X

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Global quotients: a stringy perspective

Classical perspective: Invariants That is for (X , G) look at X /G, where X /G is the space of orbits.

Stringy Perspective: Inertia variety For (X , G) study

g I (X , G) = qg∈G X

where X g = {x ∈ X : g(x) = x} is the set of g–ﬁxed points, and q is the disjoint union.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Global quotients a stringy perspective

The space I (X , G) has a natural G action We have h(X g ) ⊂ X hgh−1 . Let x ∈ X g then (hgh−1)(h(x)) = h(g((hh−1)(x))) = h(g(x)) = h(x)

Inertia stack IG

IG (X , G) := [I (X , G)/G] Let C(G) be the set of conjugacy classes of G, then

g IG (X , G) = q[g]∈C(G)[X /Z(g)]

where [g] runs through a system of representatives, and Z(g) is the centralizer of g.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Inertia stack

Structure of inertia orbifold/stack The inertia variety exists only for a global quotient, while the inertia stack can be deﬁned for a DM stack a Xf:= X(g), (g)

where the indices run over conjugacy classes of local

automorphisms, and X(g) = {(x, (g))|g ∈ Gx }/ZGx (g).

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Examples of the stringy spaces

Example 1: Cone X = C, G = C3 1 X /G = Cone 2 I (X , G) = C q Vertex q Vertex 3 IG (Y , G) = Cone q Vertex q Vertex.

Example 2: Symmetric Products 2 Y = X × X , G = S2 = h1, τ : τ = 1i. ∆ : X → X × X ; x 7→ (x, x) the diagonal.

1 X /G = {{x, y} : x, y ∈ X } the set of unordered pairs

2 I (Y , G) = I (X × X , S2) = (X × X ) q ∆(X ) 3 IG (Y , G) = {{x, y} : x, y ∈ X } q ∆(X )

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Motivation for the stringy perspective

Physics Mathematics • “Twist ﬁelds” from orbifold • Cobordisms of bordered conformal/topological ﬁeld surfaces with principal theory. G–bundles. • Orbifold Landau–Ginzburg • Orbifolded singularities with theories. symmetries. • Orbifold string theory. • Orbifold Gromov–Witten Invariants.

An open interval in X becomes a ζx closed S1 under the action of G. x [x]

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Orbifolds/Stacks Stringy Orbifolds/Stacks Motivation Mathematical Motivation

1 New methods/invariants for singular spaces. 2 The slogan: “string smoothes out singularities”. Orbifold K–Theory Conjecture: The orbifold K–Theory of (X , G) is isomorphic to the K–Theory of a certain Y , such that Y → X is a crepant resolution of singularities. 3 Mirror–Symmetry. One can expect a general construction using orbifolds as the mathematical equivalent for orbifold Landau–Ginzburg models. 4 New methods/applications for representation theory.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Stringy Multiplication

1 Given: (X , G) and a functor, which has a multiplication (E.g. ∗ ∗ ∗ K, A , Ktop, H ). Let’s fix coefficients to lie in Q. L g 2 Fact: K (X , G) := K(I (X , G)) = g∈G K(X ) is a vector ∗ ∗ ∗ space. (The same holds for H , A , Ktop) 3 Problem: From the stringy (cobordism/“twist field”) point of view one expects a stringy G–graded multiplication.

K(X g ) ⊗ K(X h) → K(X gh)

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Pushing, Pulling and Obstructions

Again, let’s fix Q as coefficients. We will first deal with the functor K. Set X hg,hi = X g ∩ X h

X g X h X gh e1 - e2 ↑ % e3 X hg,hi

∗ 1 One naturally has pull–back operations ei , since the functors are contra-variant. This is basically just the restriction.

2 On also has push–forward operations ei∗.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The Stringy Multiplication

Ansatz (basic idea) K(X g ) K(X h) K(X gh) g h ∗ ∗ For Fg ∈ K(X ), Fh ∈ K(X ) e1 & e2 ↓ % e3∗ K(X hg,hi)

∗ ∗ Fg · Fh := e3∗(e1 (Fg ) ⊗ e2 (Fh)⊗ObsK (g, h))

The Obstruction

The above formula without ObsK (g, h) will not give something associative in general. There are also natural gradings which would not be respected without ObsK (g, h) either.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The Stringy Multiplication

Ansatz (basic idea) K(X g ) K(X h) K(X gh) g h ∗ ∗ For Fg ∈ K(X ), Fh ∈ K(X ) e1 & e2 ↓ % e3∗ K(X hg,hi)

∗ ∗ Fg · Fh := e3∗(e1 (Fg ) ⊗ e2 (Fh)⊗ObsK (g, h))

The Obstruction

The above formula without ObsK (g, h) will not give something associative in general. There are also natural gradings which would not be respected without ObsK (g, h) either.

Ralph Kaufmann Stringy orbifold K-Theory (Maybe) Later.

Now. When both deﬁnitions apply, they agree.

Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The stringy multiplication

The solution ∗ ObsK(g, h) = λ−1(R(g, h) )

∗ ∗ ∗ Fg · Fh := e3∗(e1 (Fg ) ⊗ e2 (Fh)⊗λ−1(R(g, h) ))

Two sources for the obstruction bundle R(g, h) 1 R(g, h) from GW theory/mapping of curves [CR02/04,FG03,JKK05] (Initially only for H∗). 2 R(g, h) from K-theory and representation theory. [JKK07 (Inv. Math)].

Ralph Kaufmann Stringy orbifold K-Theory (Maybe) Later.

When both deﬁnitions apply, they agree.

Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The stringy multiplication

The solution ∗ ObsK(g, h) = λ−1(R(g, h) )

∗ ∗ ∗ Fg · Fh := e3∗(e1 (Fg ) ⊗ e2 (Fh)⊗λ−1(R(g, h) ))

Ralph Kaufmann Stringy orbifold K-Theory When both deﬁnitions apply, they agree.

Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The stringy multiplication

The solution ∗ ObsK(g, h) = λ−1(R(g, h) )

∗ ∗ ∗ Fg · Fh := e3∗(e1 (Fg ) ⊗ e2 (Fh)⊗λ−1(R(g, h) ))

Two sources for the obstruction bundle R(g, h) 1 R(g, h) from GW theory/mapping of curves [CR02/04,FG03,JKK05] (Initially only for H∗). (Maybe) Later. 2 R(g, h) from K-theory and representation theory. [JKK07 (Inv. Math)]. Now.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The stringy multiplication

The solution ∗ ObsK(g, h) = λ−1(R(g, h) )

∗ ∗ ∗ Fg · Fh := e3∗(e1 (Fg ) ⊗ e2 (Fh)⊗λ−1(R(g, h) ))

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The obstruction bundle

Eigenspace decomposition Let g ∈ G have the order r. hgi ⊂ G acts on X and leaves X g invariant. So hgi acts on the restriction of the tangent bundle

TX |X g and the latter decomposes into Eigenbundles Wg,k whose Eigenvalue is exp(−2πki/r) for the action of g.

r−1 M TX |X g = Wg,k k=1

Ralph Kaufmann Stringy orbifold K-Theory The Eigenvalues are 1 1 = exp(2πi0) und −1 = exp(2πi 2 )

Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Example: The symmetric product

X ∆

(x,y) v

XxX X

Ralph Kaufmann Stringy orbifold K-Theory The Eigenvalues are 1 1 = exp(2πi0) und −1 = exp(2πi 2 )

Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Example: The symmetric product

X ∆

(x,y) v −v

(y,x)

XxX X

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Example: The symmetric product

X ∆

(x,y) v −v

(y,x)

XxX X The Eigenvalues are 1 1 = exp(2πi0) und −1 = exp(2πi 2 )

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Stringy K-Theory

The bundles S r−1 M k g Sg := W ∈ K(X ) r g,k k=1

Theorem (JKK)

Let X be a smooth projective variety with an action of a ﬁnite group G, then

hg,hi (g, h) = TX TX | hg,hi ⊕ | ⊕ | hg,hi ⊕ −1 R X Sg X hg,hi Sh X S(gh) X hg,hi deﬁnes a stringy multiplication on K (X , G) (actually a categorical–G–Frobenius algebra).

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The “stringy” Chern–Character

Chow-Ring Let A be the Chow-Ring A (X , G) := A(I (X , G)). For g h vg ∈ A(X ), vh ∈ A(X ), ObsA(g, h) = ctop(R(g, h)) set

∗ ∗ vg · vh := e3∗(e1 (vg ) ⊗ e2 (vh) ⊗ ObsA(g, h))

then this deﬁnes an associative multiplication (categorical-G-Frobenius algebra).

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The “stringy” Chern–Character

Chern–Character Let X be a smooth, projective variety with an action of G. Deﬁne C h : K (X , G) → A (X , G) via

−1 C h(Fg ) := ch(Fg ) ∪ td (Sg )

g Here Fg ∈ K(X ), td is the Todd class and ch is the usual Chern–Character.

Theorem (JKK)

C h : K (X , G) → A (X , G) is an isomorphism. An isomorphism of categorical–G–Frobenius algebras.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The “stringy” Chern–Character

Chern–Character Let X be a smooth, projective variety with an action of G. Deﬁne C h : K (X , G) → A (X , G) via

−1 C h(Fg ) := ch(Fg ) ∪ td (Sg )

g Here Fg ∈ K(X ), td is the Todd class and ch is the usual Chern–Character.

Theorem (JKK)

C h : K (X , G) → A (X , G) is an isomorphism. An isomorphism of categorical–G–Frobenius algebras.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Variations

Theorems 1 If X is an equivariantly stable almost complex manifold, then ∗ analogous results hold for the topological K–theory Ktop and cohomology H∗ yielding an isomorphism of G–Frobenius algebras. 2 For a nicea stack X there are respective versions for the K–theory, the Chow Rings und and the Chern–Characters using the inertia stacks. 3 Notice the stringy K is usually “bigger” than stringy Chow and C h is only a ring homomorphism. For a global quotient (X /G) we have that K (X , G)G embeds into the full stringy K.

ae.g. X its inertia and double inertia smooth with resolution property; for instance X smooth DM with ﬁnite stabilizers. Special cases are [X /G ], where G is a Lie group swhich operates with ﬁnite stabilizers.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Variations

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Compatibility

Compatibility with the curve constructions in the case of H∗ 1 If X is a complex manifold, then the multiplication on H∗ coincides with that deﬁned by Fantechi–Goettsche on H∗(I (X , G)). 2 In the orbifold case, i.e. [X /G ] as above, the multiplication on H∗ coincides with that deﬁned by Chen–Ruan.

Orbifold K-Theory Conjecture ×n [n] For a K3–surface X , Y = (X , Sn) and Z = Hilb (X ) as its resolution Z → Y the Orbifold K-theory conjecture holds, with a concrete choice of discrete torsion.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Simple examples

pt/G Let pt be a point with a trivial operation of G and let’s take coeﬃcients in C: ∀g ∈ G : g(pt) = pt. Now, I (pt, G) = qg∈G pt and ∗ L H (pt, G) = g∈G C ' C[G]. The G–invariants are the class functions. Symmetric Products (Second quantization [K04]) Let X be a smooth projective variety. The diagonal ∆ : X → X × X ; x 7→ (x, x) deﬁnes a push–forward ∗ ∗ ∗ ∗ ∆∗ : H (X ) → H (X × X ) ' H (X ) ⊗ H (X ) ∗ ∗ ∗ ∗ H (X × X , S2) = (H (X ) ⊗ H (X )) ⊕ H (X ) ∗ ∗ with the multiplication xτ · yτ = x ⊗ y · ∆∗(1) ∈ H (X ) ⊗ H (X ).

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Frobenius Algebras

Definition A Frobenius algebra is a finite dimensional, commutative, associative unital algebra R with a non–degenerate symmetric paring h , i which satisfies hab, ci = ha, bci Example H∗(X , k) for X a compact oriented manifold or a smooth projective smooth variety. The pairing is the Poincaréparing which top is essentially given by the integral over X . Notice over Q, K and H∗ are isomorphic, but not isometric (Hirzebruch-Riemann-Roch).

Notice that if Lv is the left multiplication by v, there is a trace τ : R → k: τ(v) := Tr(Lv )

Ralph Kaufmann Stringy orbifold K-Theory Remark For these examples we could also use an alternative deﬁnition of a Frobenius object, which is given as an algebra µ and a co-algebra ∆ with unit η and co-unit which satisfy ∆ ◦ µ = (µ ⊗ id) ◦ (id ⊗ ∆) Then τ(v) = (µ(v, ∆ ◦ η(1)). This ﬁts well with TFT.

Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Pre-Frobenius algebras

Ralph Kaufmann Stringy orbifold K-Theory This ﬁts well with TFT.

Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Pre-Frobenius algebras

Deﬁnition A pre–Frobenius algebra is a commutative, associative unital algebra R together with a trace element τ : R → k. Example A∗(X ) and K ∗(X ) for a smooth projective variety. The trace element is given by the integral and Euler characteristic respectively. Remark For these examples we could also use an alternative deﬁnition of a Frobenius object, which is given as an algebra µ and a co-algebra ∆ with unit η and co-unit which satisfy ∆ ◦ µ = (µ ⊗ id) ◦ (id ⊗ ∆) Then τ(v) = (µ(v, ∆ ◦ η(1)).

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Pre-Frobenius algebras

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW G–Frobenius algebras

Definition (concise form) [K-Pham] A strict categorical G–Frobenius algebra is a Frobenius objects in the braided monoidal category of modules over the quasi–triangular quasi–Hopf algebra defined by the Drinfeld double of the group ring D(k[G]) which satisfies two additional axioms: T Invariance of the twisted sectors. S The trace axiom.

Pre– and Non–Strict • There is a notion of pre-G–FA, which uses trace elements which satisfy T as extra data. • Non–strict, means that there are certain characters which appear. (This is needed in Singularity Theory. Today only strict).

Ralph Kaufmann Stringy orbifold K-Theory ; discrete torsion [K04]

Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW More details/examples

Some details/consequences If R is a GFA then L 1 R = g∈G Rg is a G–graded algebra 2 R has a G action ρ such that ρ(g)(Rh) ⊂ Rghg −1 (Drinfel’d–Yetter)

3 ag bh = ρ(g)(bh)ag (twisted commutativity)

4 (T) g|Rg = id −1 5 (S) If v ∈ R[g,h] TrRh (ρ(g ) ◦ Lv ) = TrRg (Lv ◦ ρ(h))

Examples k[G] = H∗(pt, G) and kα[G] for α ∈ Z 2(G, k∗).

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW More details/examples

Some details/consequences If R is a GFA then L 1 R = g∈G Rg is a G–graded algebra 2 R has a G action ρ such that ρ(g)(Rh) ⊂ Rghg −1 (Drinfel’d–Yetter)

3 ag bh = ρ(g)(bh)ag (twisted commutativity)

4 (T) g|Rg = id −1 5 (S) If v ∈ R[g,h] TrRh (ρ(g ) ◦ Lv ) = TrRg (Lv ◦ ρ(h))

Examples k[G] = H∗(pt, G) and kα[G] for α ∈ Z 2(G, k∗). ; discrete torsion [K04]

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Quantum and/or stringy

Gromov-Witten invariants of a variety V [Ruan-Tian, Kontsevich-Manin, Behrend-Fantechi, Li-Tian, Siebert, ...]

evi Mg,n(V , β) −−−−→ V   y

Mg,n ∗ lead to operations on cohomology, for ∆i ∈ H (V ) Z Y ∗ < ∆1,..., ∆n >:= evi (∆i ) virt [Mg,n]

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Quantum and/or stringy

Quantum K-theory [Givental-Lee] Realization of Givental:

virt “[Mg,n] = ctop(Obs)” in good cases ; quantum K-theory: Operations on K(V )

Y ∗ ∗ < F1,..., Fn >:= χ( evi (Fi )λ−1(Obs ))

in both cases get classical Frobenius algebra for n = 3, β = 0. The other operations give a deformation, viz. Frobenius manifold or CohFT .

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Orbifold GW theory, stringy mutliplication

Orbifold GW theory [Chen-Ruan] n X orbifold (locally R /G where G ﬁnite group). They constructed moduli spaces of orbifold stable maps to get operations on ∗ ∗ HCR (X ) = H (I )G (X , G)) where IG (X , G) is the inertia orbifold or stack. New classical limit (n = 3, β = 0) plus new “stringy” ring structure.

Algebraic setting [Abramovich-Graber-Vistoli] X a DM-stack over a ﬁeld of characteristic zero.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW Global orbifolds

Global orbifold cohomology I [Fantechi-Göttsche] For Y = X /G with G finite defined a ring structure and G-action ∗ L ∗ g ∗ G ∗ on HFG (X ) = g∈G H (X ) such that HFG (X ) ' HCR (X ).

Global orbifold cohomology II [Jarvis, K, Kimura] G G Gave a construction for the spaces Mg,n and Mg,n(V , β), constructed the virtual fundamental class for β = 0 (and all β for a ∗ trivial action) and showed that HJKK (X ) is a G-Frobenius algebra ∗ ∗ and that HJKK (X ) ' HFG (X ). Also gave analog of Frobenius manifold deformation viz. G − CohFT and showed that G-invariants of a G − CohFT are a CohFT , i.e. a Frobenius manifold.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Stringy Functors Push–Pull Stringy K Thms Algebra GW The obstruction bundle: curve case

The Obstruction bundle II 1 The principal hmi-bundle over P − {0, 1, ∞} with monodromies mi extends to a smooth connected curve E. E/hm1, m2i has genus zero, and the natural map E → E/hmi is branched at the three points p1, p2, p3 with monodromy m1, m2, m3, respectively. Let π : E × X hm1,m2i → X hm1,m2i be the second projection. We deﬁne hm ,m i the obstruction bundle R(m1, m2) on X 1 2 to be

1 hm1,m2i R(m1, m2) := R π∗ (OE TX |X hm1,m2i ).

Theorem (JKK) When both deﬁnitions of R make sense, they agree.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Summary

Constructions ∗ ∗ ∗ ∗ We have stringy versions for the functors A , K , H , Ktop and Chern characters, in two settings 1 Global Orbifolds. C h is a ring isomorphism of G-FAs. 2 Nice stacks. C h is a ring homomorphism. (K carries more data). These results hold in 1 In the algebraic category 2 In the stable almost complex category

Features We can deﬁne these without reference to moduli spaces of maps. Get examples of the crepant resolution conjecture.

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Newer Developments, based on our formalism

A slew of activities 1 Orbifold deRham Theory (non–Abelian case) [K07]. 2 Hochschild, S1 equivariant version [Pﬂaum et al. 07]. 3 Stringy Singularity Theory [K,Libgober in prep]. 4 Symplectic theory [Goldin, Holm, Knudsen, Harada, Kimura, Matusumura 07/09]. 5 Equivariant versions [Jarvis Kimura Edidin 09] [K Edidin]. 6 Stringy orbifold string topology [Gonz´aleset al.]. 7 Gerbe Twists [Adem, Ruan, Zhang 06] 8 Global Gerbe twists using twisted Drinfel’d double [K, Pham 08] 9 Higher twists [Pham 09] 10 Wreath products [Matsumura 06]

Ralph Kaufmann Stringy orbifold K-Theory Thanks!!!

Outline Intro Stringy K–Theory Summary & Future Still more to come, so check back ...

Ralph Kaufmann Stringy orbifold K-Theory Outline Intro Stringy K–Theory Summary & Future Still more to come, so check back ...

Thanks!!!

Ralph Kaufmann Stringy orbifold K-Theory